# Bayesian Estimation and Tracking: A Practical Guide

ISBN: 978-0-470-62170-7
400 pages
June 2012

## Description

A practical approach to estimating and tracking dynamic systems in real-worl applications

Much of the literature on performing estimation for non-Gaussian systems is short on practical methodology, while Gaussian methods often lack a cohesive derivation. Bayesian Estimation and Tracking addresses the gap in the field on both accounts, providing readers with a comprehensive overview of methods for estimating both linear and nonlinear dynamic systems driven by Gaussian and non-Gaussian noices.

Featuring a unified approach to Bayesian estimation and tracking, the book emphasizes the derivation of all tracking algorithms within a Bayesian framework and describes effective numerical methods for evaluating density-weighted integrals, including linear and nonlinear Kalman filters for Gaussian-weighted integrals and particle filters for non-Gaussian cases. The author first emphasizes detailed derivations from first principles of eeach estimation method and goes on to use illustrative and detailed step-by-step instructions for each method that makes coding of the tracking filter simple and easy to understand.

Case studies are employed to showcase applications of the discussed topics. In addition, the book supplies block diagrams for each algorithm, allowing readers to develop their own MATLAB® toolbox of estimation methods.

Bayesian Estimation and Tracking is an excellent book for courses on estimation and tracking methods at the graduate level. The book also serves as a valuable reference for research scientists, mathematicians, and engineers seeking a deeper understanding of the topics.

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Preface

Acknowledgments

List of Figures xi

List of Tables xxi

Part I. Prelininaries

1. Introduction 3

1.1 Bayesian Inference 5

1.2 Bayesian Hierarchy of Estimation Methods 7

1.3 Scope of this Text 8

1.4 Modeling and Simulation with Matlab® 13

2. Preliminary Mathematical Concepts 19

2.1 A Very Brief Overview of Matrix Linear Algebra 20

2.2 Vector Point Generators 27

2.3 Approximating Nonlinear Multidimensional Functions with Multidimensional Arguments 32

2.4 Overview of Multivariate Statistics 47

3. General Concepts of Bayesian Estimation 69

3.1 Bayesian Estimation 70

3.2 Point Estimators 72

3.3 Introduction to Recursive Bayesian Filtering of Probability Density Functions 76

3.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance 81

3.5 Discussion of General Estimation Methods 88

4. Case Studies: Preliminary Discussions 93

4.1 The Overall Simulation/Estimation/Evaluation Process 94

4.2 A Scenario Simulator for Tracking a Constant-Velocity Target Through a DIFAR Buoy Field 97

4.3 DIFAR Buoy Signal Processing 102

4.4 The DIFAR Likelihood Function 111

Part II. The Gaussian Assumption: A Family of Kalman Filter Estimators

5. The Gaussian Noise Case: Multidimensional Integration of Gaussian-Weighted Distributions 119

5.1 Summary of Important Results From Chapter 3 122

5.2 Derivation of the Kalman Filter Correction (Update) Equations Revisted 124

5.3 The General Bayesian Point Prediction Integrals for Gaussian Densities 128

6. The Linear Class of Kalman Filters 141

6.1 Linear Dynamic Models 142

6.2 Linear Observation Models 143

6.3 The Linear Kalman Filter 144

6.4 Application of the LKF to DIFAR Buoy Bearing Estimation 146

7. The Analytical Linearization Class of Kalman Filters: The Extended Kalman Filter 153

7.1 One-Dimensional Consideration 154

7.2 Multidimensional Consideration 159

7.3 An Alternate Derivation of the Multidimensional Covariance Prediction Equations 172

7.4 Application of the EKF to the DIFAR Ship Tracking Case Study 174

8. The Sigma Point Class: The Finite Difference Kalman Filter 187

8.1 One-Dimensional Finite Difference Kalman Filter 189

8.2 Multidimensional Finite Difference Kalman Filters 195

8.3 An Alternate Derivation of the Multidimensional Finite Difference Covariance Prediction Equations 201

9. The Sigma Point Class: The Unscented Kalman Filter 207

9.1 Introduction to Monomial Cubature Integration Rules 207

9.2 The Unscented Kalman Filter 211

9.3 Applications of the UKF to the DIFAR Ship Tracking Case Study 221

10. The Sigma Point Class: The Spherical Simplex Kalman Filter 227

10.1 One-Dimensional Spherical Simplex Sigma Points 228

10.2 Two-Dimensional Spherical Simplex Sigma Points 229

10.3 Higher-Dimensional Spherical Simplex Sigma Points 233

10.4 The Spherical Simplex Kalman Filter 233

10.5 The Spherical Simplex Kalman Filter Process 236

10.6 Application of the SSKF to the DIFAR Ship Tracking Case Study 236

11. The Sigma Point Class: The Gauss-Hermite Kalman Filter 241

11.2 One-Dimensional Gauss-Hermite Kalman Filter 248

11.3 Multidimensional Gauss-Hermite Kalman Filter 251

11.4 Sparse Grid Approximation for High Dimension/High Polynomial Order 257

11.5 Application of the GHKF to the DIFAR Ship Tracking Case Study 261

12. The Monte Carlo Kalman Filter 265

12.1 The Monte Carlo Kalman Filter 268

13. Summary of Gaussian Kalman Filters 273

13.1 Analytical Kalman Filters 274

13.2 Sigma-Point Kalman Filters 276

13.3 A More Practical Approach to Utilizing the Family of Kalman Filters 284

14. Performance Measures for the Family of Kalman Filters 289

14.1 Error Ellipses 290

14.2 Root Mean Squared Errors 299

14.3 Divergent Tracks 301

14.4 Cramer-Rao Lower Bound 302

14.5 Performance of Kalman Class DIFAR Track Estimators 315

Part III. Monte Carlo Methods

15. Introduction to Monte Carlo Methods 323

15.1 Approximating a Density From a Set of Monte Carlo Samples 325

15.2 General Concepts Importance Sampling 340

15.3 Summary 347

16. Sequential Importance Sampling Particle Filters 351

16.1 General Concept of Sequential Importance Sampling 351

16.2 Resampling and Regularization (Move) for SIS Particle Filters 357

16.3 The Bootstrap Particle Filter 372

16.4 The Optimal SIS Particle Filter 378

16.5 The SIS Auxiliary Particle Filter 385

16.6 Approximations to the SIS Auxiliary Particle Filter 393

16.7 Reducing the Computational Load Through Rao-Blackwellization 396

17. The Generalized Sequential Monte Carlo Particle Filter 403

17.1 The Gaussian Particle Filter 404

17.2 The Combination Particle Filter 406

17.3 Performance Comparison of all DIFAR Tracking Filters 411

18. A Spherical Constant Velocity Model for Target Tracking in Three Dimensions 421

18.1 Tracking a Target in Cartesian Coordinates 426

18.2 Tracking a Target in Spherical Coordinates 433

18.3 Implementation of Cartesian and Spherical Tracking Filters 443

18.4 Performance Comparison for Various Estimation Methods 453

18.5 Some Observations and Future Considerations 469

19. Tracking a Falling Rigid Body Using Photogrammetry 497

19.1 Introduction 497

19.2 The Process (Dynamic) Model for Rigid Body Motion 502

19.3 Components of the Observation Model 513

19.4 Estimation Methods 517

19.5 The Generation of Synthetic Data 529

19.6 Performance Comparison Analysis 538

20. Sensor Fusion using Photogrammetric and Inertial Measurements 559

20.1 Introduction 559

20.2 The Process (Dynamic) Model for Rigid Body Motion 562

20.3 The Sensor Fusion Observational Model563

20.4 The Generation of Synthetic Data 569

20.5 Estimation Methods 572

20.6 Performance Comparison Analysis 577

20.7 Conclusions 585

20.8 Future Work 586

References 589

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## Author Information

ANTON J. HAUG, PhD, is member of the technical staff at the Applied Physics Laboratory at The Johns Hopkins University, where he develops advanced target tracking methods in support of the Air and Missile Defense Department. Throughout his career, Dr. Haug has worked across diverse areas such as target tracking; signal and array processing and processor design; active and passive radar and sonar design; digital communications and coding theory; and time- frequency analysis.

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